APPM 3310: Matrix Methods — Exam #1 — October 3, 2007
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Show all work in your bluebook. A correct answer with no supporting work may receive no
credit while an incorrect answer with some correct work may receive partial credit. No electronic
devices of any kind (e.g. cell phones, calculators, etc.) are permitted.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
1. (10 points) Compute the determinant of matrix Bin terms of λ,b1,b2,b3and b4.
B=
λ−1 0 0
0λ−1 0
0 0 λ−1
b1b2b3b4+λ
2. (40 points) Let Abe an m×nmatrix.
(a) Give the definitions for the range of Aand the corange of A.
(b) State the Fundamental Theorem of Linear Algebra.
(c) Now find a basis and the dimension for the range, corange, and kernel of the matrix
A=
123
456
789
(d) Use the matrix Ain part (c) to give conditions on b=
b1
b2
b3
such that Ax=bhas a
solution.
(e) Restate your answer in (d) using a subspace you found in part (c). (This should be no more
than one or two sentences.)
3. (30 points) Let P(2) be the set of polynomials of degree less than or equal to 2.
(a) Show that P(2) is a subspace of the vector space of all polynomials.
(b) Give the definition of linear independence. Are the polynomials p1= 1, p2=x, and
p3= (−1+3x2)/2 linearly independent?
(c) Give the definition of a basis. Are the polynomials in part (b) a basis for P(2) ? Explain
briefly.
4. (30 points)
(a) Let u= (2,1,1)Tand v= (1,0,1)T. Compute A=I−uvTand find A−1.
(b) Now, let uand vbe arbitrary n×1 vectors with vTu6= 1. Let A=I−uvT. Give the
definition for the inverse of a matrix and use it to verify that A−1=I+1
1−vTuuvT.
(Hint: Remember that vTuis a scalar and uvTis a matrix.)